Integration by Parts Rule

What is Integration?

  • In Mathematics, when general operations like addition operations cannot be performed, we use integration to add values on a large scale.

  • There are different types of methods in mathematics to integrate functions.

  •  Integration and differentiation are also a pair of inverse functions similar to  addition - subtraction, and multiplication-division. 

  • The process of finding functions whose derivative is given is named anti - differentiation or integration.

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\[\int_{a}^{b} f(x) dx\] = value of the anti-derivative at upper limit b – the value of the same anti-derivative at lower limit a.


Here’s What Integration is!

If \[\frac{d}{dx} (F(x)) = f(x)\], then


\[\int f(x)dx = F(x) + c\]


The function F(x) is called anti-derivative or integral or primitive of the given function f(x) and c is known as the constant of integration or the arbitrary constant.


The function f(x) is called the integrand and f(x)dx is known as the element of integration.


Points to Remember

  • Since the integral of a function isn’t definite, therefore it is generally referred to as indefinite integral.

  • We can never find the integral of a function at a point; we always find the integral of a given function in an interval.

  • Integral of a function is not unique; integrals of a function differ by numbers.


Types of Integration Maths or the Integration Techniques

Here’s a list of Integration Methods –

  1. Integration by Substitution

  2. Integration by Parts Rule

  3. Integration by Partial Fraction

  4. Integration of Some particular fraction

  5. Integration Using Trigonometric Identities

In this article we are going to discuss the integration by parts rule, integration by parts formula, integration by parts examples and integration by parts examples and solutions.


Integration by Parts Rule

  • If the integrand function can be represented as a multiple of two or more functions, the integration of any given function can be done by using the Integration by Parts rule.

  • Let us take an integrand function which is equal to u(x)v(x).

  • In mathematics, Integration by part basically uses the ILATE rule that helps to select the first function and second function in the integration by parts method.

  • Integration by parts formula,

            ∫u(x).v(x).dx = u(x).∫v(x).dx–∫(u′(x).∫v(x).dx).dx

The integration by parts formula, can be further written as integral of the product of any two functions = (First function × Integral of the second function) – Integral of [ (differentiation of the first function) × Integral of the second function]


From the Integration by parts formula discussed above,

  • u is the function u(x)

  • v is the function v(x)

  • u' is the derivative of the function u(x)

As a diagram:


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Ilate Rule

In integration by parts, we have learned when the product of two functions are given to us then we apply the required formula. The integral of the two functions is taken, by considering the left term as first function and second term as the second function. This method is called Ilate rule. Suppose, we have to integrate xex, then we consider x as the first function and ex as the second function. So basically, the first function is chosen in such a way that the derivative of the function could be easily integrated. Usually, the preference order of this rule is based on some functions such as Inverse, Algebraic, Logarithm, Trigonometric, Exponent. This rule helps us to solve integration by parts examples using Integration by parts formula.


What is the ILATE rule?


ILATE is a rule which helps to decide which term should you differentiate first and which term should you integrate first.

  • I-Inverse

  • L-Logarithmic

  • Algebraic

  • T-Trigonometric

  • E-Exponential

The term which is closer to I is differentiated first and the term which is closer to E is integrated first.


Note

  • Integration by parts rule is not applicable for functions such as ∫ √x sin x dx.

  • We do not add any constant while finding the integral of the second function.

  • Usually, if any function is a power of x or a polynomial in x, then we take it as the first function. However, in cases where another function is an inverse trigonometric function or logarithmic function, then we take them as the first function.


Rules to be Followed for Solving Integration by Parts Examples:

So we followed these steps:

  • Choose u and v functions

  • Differentiate u: u'

  • Integrate v: ∫v dx

  • Put u, u' and put ∫v dx into the given formula: u∫v dx −∫u' (∫v dx) dx

  • Simplify and solve the Integration by parts examples

In simpler words, to help you remember,the following ∫u v dx becomes:

(u integral v) - integral of (derivative u, integral v)


Standard Integrals in Integration

\[\int x^{n} dx\]

\[\frac{x^{n + 1}}{n + 1}\] + C, where n \[\neq\] -1.

\[\int {\text{sinx dx}}\]

- cos x + C

\[\int {\text{cosx dx}}\]

sin x + C

\[\int sec^{2}x dx\]

tan x +C

\[\int cosec^{2}x dx\]

- cot x + C

\[\int {\text{secx tanx dx}}\]

sec x + C

\[\int {\text{cosecx cotx dx}}\]

- cosec x + C


Let’s understand better by solving integration by parts examples and solutions.


Questions to be Solved

Question 1) What is ∫x cos(x) dx ?

Answer)We have x multiplied by cos(x), so integration by parts is a good choice.

First choose which functions for u and v:

  • u = x

  • v = cos(x)

So now, we have obtained it in the format ∫u v dx and we can proceed:

Differentiate u: u' = x' = 1

Integrate the v part : ∫v dx = ∫cos(x) dx = sin(x)

Now we can put it together and we get the answer:


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FAQ (Frequently Asked Questions)

Question 1)What is Integration by Parts Formula?

Answer)The method Integration by Parts is known to be a special method of integration that is often useful. We use it when two functions are multiplied together, but are also helpful in many other ways. Let us see the rule of integration by parts: ∫u v dx equals u∫v dx −∫u' (∫v dx) dx. u is the function u(x) is the formula for Integration by Parts.

Question 2)Can you use Integration by Parts on any Integral?

Answer)Yes, we can use integration by parts to integrate any function. But the real problem is that we want integration by parts to be used instead of a substitution method for every function in integration. And some functions can only be integrated using the integration by parts method like ln(x).

Question 3)What is the Product Rule of Integration?

Answer)The Product Rule of Integration enables you to integrate the product of two functions. For example, through a series of mathematical somersaults, you can turn the following equation into a formula that's useful for integrating.

Question 4)When Should I use Integration by Parts?

Answer)Integration by parts is for functions that can be written as the product of another function and a third function's derivative. A good rule of thumb to follow would be to try the u-substitution first, and then if you cannot reformulate your function into the correct form, then you can try integration by parts.